Abstract: In the context of a star cluster moving on a circular galactic orbit, a ‘potential escaper’ is a cluster star that has orbital energy greater than the escape energy, and yet is confined within the Jacobi radius of the stellar system. On the other hand, analytic models of stellar clusters typically have a truncation energy equal to the cluster escape energy, and therefore explicitly exclude these energetically unbound stars. Starting from the landmark analysis performed by He ́non of periodic orbits of the circular Hill equations, we present a numerical exploration of the population of ‘non-escapers’, defined here as those stars that remain within two Jacobi radii for several galactic periods, with energy above the escape energy. We show that they can be characterized by the Jacobi integral and two further approximate integrals, which are based on perturbation theory and ideas drawn from Lidov–Kozai theory. Finally, we us e these results to construct an approximate analytic model that includes a phase-space description of a population resembling that of potential escapers, in addition to the usual bound population.